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Brazilian Journal
of Chemical
Engineering
Vol. 36, No. 01, pp. 421  437, January  March, 2019
dx.doi.org/10.1590/01046632.20190361s20170379
STATE ESTIMATION AND TRAJECTORY
TRACKING CONTROL FOR A NONLINEAR
AND MULTIVARIABLE BIOETHANOL
PRODUCTION SYSTEM
* Corresponding author: M. Cecilia Fernández  Email: mcfernandez@unsj.edu.ar
M. Cecilia Fernández1*, M. Nadia Pantano1, Francisco G. Rossomando2,
O. Alberto Ortiz1 and Gustavo J. E. Scaglia1
1 Instituto de Ingeniería Química, Universidad Nacional de San Juan, Consejo Nacional de Investigaciones Cientícas y Técnicas,
San Juan J5400ARL, Argentina. Email: mcfernandez@unsj.edu.ar, ORCID: 0000000323120163; npantano@unsj.edu.ar,
ORCID: 0000000325496535; rortiz@unsj.edu.ar, ORCID: 0000000266609674; gscaglia@unsj.edu.ar, ORCID: 0000000201880017
2 Instituto de Automática, Universidad Nacional de San Juan, Consejo Nacional de Investigaciones Cientícas y Técnicas,
San Juan J5400ARL, Argentina. Email: frosoma@inaut.unsj.edu.ar, ORCID: 0000000277928101
(Submitted: July 17, 2017 ; Revised: August 8, 2018 ; Accepted: August 13, 2018)
Abstract  In this paper a controller is proposed based on linear algebra for a fedbatch bioethanol production
process. It involves nding feed rate proles (control actions obtained as a solution of a linear equations system)
in order to make the system follow predened concentration proles. A neural network states estimation is
designed in order to know those variables that cannot be measured. The controller is tuned using a Monte Carlo
experiment for which a cost function that penalizes tracking errors is dened. Moreover, several tests (adding
parametric uncertainty and perturbations in the control action) are carried out so as to evaluate the controller
performance. A comparison with another controller is made. The demonstration of the error convergence, as
well as the stability analysis of the neural network, are included.
Keywords: Fedbatch bioprocess; Nonlinear and multivariable system; Proles tracking control; Numerical
methods/linear algebra; State estimation.
INTRODUCTION
In recent years, the bioprocess industry, so called
white biotechnology (Heux et al., 2015), has gained an
important position. This is because it has an important
role in the production of highadded value products;
such as recombinant proteins, vaccines and antibiotics
in the pharmaceutical industry, or beer, wine, yeast in
the manufacturing of agrofood goods, or biogas and
compost in the treatment of urban and industrial solid
organic wastes and wastewater (Mangesh and Jana,
2008), between others.
Alcoholic fermentation is an ancient practice,
commonly used in the production of alcoholic
beverages such as beer, wine, cider, sake and distilled
drinks (Wood, 2012). Moreover, research on using
ethanol as an alternative fuel has gained tremendous
attention all over the world since the petroleum crisis
in the 1970s (Ajbar and Ali, 2017). It is noteworthy
that all these processes are carried out in bioreactors.
There are many operating modes for bioreactors:
batch, fedbatch and continuous. Among the different
modes, the fedbatch is preferred because of the
operational exibility that it provides (Mangesh and
Jana, 2008). It consists of changing the feed rate of
nutrients, inhibitors, catalysts or inducers along the
process, whereas cells and products remain in the
fermenter until the operation ends (Hecklau et al.,
2016). Moreover, this cultivation technique has several
advantages over other operation modes (Liu et al.,
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M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
422
2013). Also, it avoids formation of unwanted products
and microorganism inhibition caused by overfeeding,
and prevents the microorganism starvation induced by
underfeeding (Ochoa, 2016). Ethanol production in a
fedbatch fermenter is common in white biotechnology
(Mangesh and Jana, 2008).
One of the main benets of fedbatch processes is
that the concentration of substrate in the cultivation
medium can be externally regulated with a suitable
feed rate prole (Liu et al., 2013), in order to obtain
better production yields (Jin et al., 2014). For this
reason, many efforts have focused on bioprocess
optimization and control so as to minimize the
production costs while increasing the yield and
productivity (Ochoa, 2016, Pantano et al., 2017a).
Nevertheless, that is an arduous task because usually
microorganisms have a complex dynamic behavior
(nonlinear and sometimes unstable) that leads to
strong modeling approximations; furthermore, there
are other complications like the presence of numerous
external disturbances or the difculty of measuring
representative variables. These problems avoid the
possibility of using PI, PID or other classic industrial
controller, which means that the development of
an specic control algorithm is necessary for each
bioprocess (De Battista et al., 2012).
The worldwide market requirements for high
standard products, as well as safe and environmentally
friendly processes, force chemical industries to
look for optimization methods and controllers that
can afford nonlinearities and transient behavior
of chemical and biochemical processes (Fujiki et
al., 2009). On the one hand, many scientists have
established optimization methods to nd the best
feed rate prole for different processes involving
fedbatch fermentations (for some examples see:
Ye et al., 2014; Kookos, 2004; del RioChanona et
al., 2016 and Dai et al., 2014, as this issue is beyond
the aim of this paper). However, while nding the
optimal feed policy of a determined process is not
an easy task, once it is known a bigger challenge is
presented: trying to track the optimal proles and
obtain repeatability from one batch to another. Recent
papers have been focused on solving this problem,
such as (Chang et al., 2016), (Jin et al., 2014), (Fu
and Chai, 2007), (Lehouche et al., 2012), (Hofmann
et al., 2017), and (Mohammadzaheri and Chen, 2011,
Hofmann et al., 2017).
Some advanced control techniques for
fermentation processes like: online adaptive control
(Guay et al., 2004), optimal control (Logist et al.,
2010), fuzzy control (Karakuzu et al., 2006), neural
networks (Imtiaz et al., 2013) and predictive control
based on nonlinear model (Preuβ et al., 2003) have
gained popularity because of their strong capability
in dealing with process nonlinearity, dynamics and
optimization. This last (NMPC), although it has
been successfully applied in numerous practical
applications, encounters a lot of limitations such
as: difculties in building accurate dynamic
models, complexity of online implementation, the
insufcient accuracy of online solutions and the
computational time required to nd the solution,
limiting its applications to bioprocesses (Jin et al.,
2014). Therefore, probably the application of the
other techniques can be difcult or even infeasible
for bioprocess. For all this reasons, the PID controller
is still the most widely used in factories (Imtiaz et
al., 2014). PID controllers are chosen mainly for
their simple structure and ease of adjustment of their
parameters. On the other hand, a major challenge is
the control set point variations, avoiding oscillations
and delays, trying to reduce tracking errors at the
same time. Therefore, although PID controllers
are sufcient to solve the control problem of many
applications in industry, for bioprocesses they do not
give good results with traditional tuning procedures.
Few works in this eld have claimed satisfactory
results tuning the parameters based on alternative
approaches such as: neural networks (Andrášik et
al., 2004) and metaheuristic algorithms (Roeva and
Slavov, 2012) to deal with the inherent complex
behavior of bioprocess.
The aim of this paper is to present a controller
for application in bioethanol production. The system
is represented with a nonlinear and multivariable
model. The controller consists of obtaining a feed
rate prole (control action) in order to track optimal
concentration proles (desired variables). The main
innovation is the simultaneously excellent tracking
of four time varying proles with only one control
action. The controller has a fast and easy design
because it allows obtaining the control action as a
solution of linear equations, even though the original
system model is nonlinear. Thus, only algebra is
needed to understand and apply this methodology. As
the controller structure comes from the mathematical
model of the process, it can be implemented in many
systems. Besides, it is versatile against different
changes and disturbances in the process and system.
This last assertion is conrmed with different tests and,
in each one, it can be observed that the error remained
at low levels (the mathematical demonstration of this
fact is also presented in Appendix A). Additionally,
the controller parameters are selected with a Monte
Carlo Randomized algorithm and a comparison with
a PID controller is shown.
In order to achieve the proposed objective, it is
assumed that the bioethanol process is well represented
by the mathematical model, the desired proles are
known and the control action can be obtained at each
instant of time. Furthermore, the state variables must
be able to be measured online, but as is known, the
states of some systems are immeasurable. Accordingly,
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
423
the estimation of those states plays a signicant role
in control design and system identication (Xu et al.,
2017). Therefore, a neural network states estimation
is designed. Fig. 1 shows a owchart of the proposed
procedure.
The paper is ordered as follows. In the next
section the process and the mathematical model
that governs it are described. The controller design
is then described; this includes the determination of
the optimal parameters of the controller, which are
obtained with a Monte Carlo experiment. Section
4 shows the neural network states estimator. The
subsequent section is organized in three different
subsections; rst, a simulation of the process under
normal operation conditions is carried out; after that,
as a demonstration of the good performance, two
tests are executed: addition of parametric uncertainty
and disturbances in the control action. Then, the
subsequent section shows a comparison with a
typical PID controller used in the industry. Finally,
the conclusions are presented.
Here, the state variables are: biomass (X), glucose
(S1), fructose (S2), ethanol (P1) and glycerol (P2)
concentrations inside the reactor. The manipulated
variable is the substrate feed rate (U). Furthermore, µ1
and µ2 represent the specic yeast cells growth rate,
qS1/P1 and qS2/P1 the specic ethanol production rate, and
qS1/P2 and qS2/P2 the specic glycerol production rate, in
all cases from glucose and fructose, respectively. V
is the culture volume. All these relations describe the
metabolic activity of microorganisms.
The variables initial values are shown in Table 1.
The parameter nomenclature, description and values
are in Table 2.
Figure 1. Flowchart of the developed technique.
Xt µµXU
VX
St q
Y
q
Y
SP
PS
SP
PS
()
=+
()
−
()
=− +
12
1
11
11
12
21
/
/
/
/ +−
()
()
=− +
+
XU
VSS
St q
Y
q
YXU
f
SP
PS
SP
PS
λ1
2
21
12
22
22
/
/
/
/VV
SS
Pt qqXU
VP
Pt qq
f
SP SP
SP S
12
11
2
11 21
12
−
()
−
()
()
=+
()
−
()
=+
λ
//
/222 2/P XU
VP
()
−
PROCESS DESCRIPTION.
In this study, an ethanol fermentation process
carried out in a fedbatch reactor is considered. The
microorganism used is Saccharomyces diastaticus.
This yeast has high ethanol tolerance and can
utilize glucose and fructose to produce ethanol
and glycerol simultaneously (Hunag et al., 2012).
The mathematical modeling of this system was
proposed by Hunag and coworkers (Hunag et al.,
2012). They used a batch fermenter to generate time
series data, which were then applied to estimate the
kinetic model parameters. After that, they applied
them to a fedbatch process to determine an optimal
control policy. This system has one input, namely,
the feed rate, and various outputs (whose proles
must follow determined variation over time), i.e.,
the cells, ethanol, glycerol, glucose and fructose
concentrations inside the reactor. The reactor’s feed
consists of a 5050 mixture of glucose and fructose.
The developed mathematical model is:
where:
Vt U
()
= q
S
KS
k
kP
SP
SP
SP
SP
SP
11
11
11
11
11
1
11
/=
++
ν
µ11
1
11
2
11
2
1
11
1
11
2
=++ ++
µm
SS
P
PP
S
S
KSSK
K
KPPK q
II
//
//
/
P
SP
SP
SP
SP
m
SS
P
S
KS
k
kP
S
KSSK
K
I
2
22
22
22
22
2
22
2
22
2
2
22
2
=
++
=++
ν
µµ22
22
12
12
12
12
11
2
1
1
KPPK qS
KS
k
k
PP
SP
SP
SP
SP
S
I
++ =+//
ν
112 2P P+
q S
KS
k
SP
SP
SP
S
21
21
21
2
2
2
/=+
νPP
SP
kP
1
21
1
+
Table 1. Initial variable values for ethanol fermentation
(Hunag et al., 2012).
CONTROLLER DESIGN
Controller Structure
Generally, the structure of a controller is constructed
based on a mathematical model. This kind of model
gives a scheme of the process from inputs to outputs.
Obviously, its quality depends on how closely those
(1)
(2)
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
424
Table 2. Parameters nomenclature, description and values (Hunag et al., 2012).
Figure 2. Cells, ethanol, glycerol and fructose reference concentrations along the process. Reference feed ow rate.
outputs match the actual process. Furthermore, it is
important to highlight that a model that accurately
replicates a real process will never be created (Zhou
et al., 1996). For this reason, a controller that allows
monitoring predened proles (obtained with a specic
model and determined condition) in the presence of
different perturbations, with minimal error, is necessary.
Below, the main contribution of this work, a control
methodology based on linear algebra for tracking
predened proles in a bioreactor is presented. For its
development the mathematical model of the process
(shown in Section 2) and the reference proles that
are wanted to be followed by the system are required.
Those proles are the concentrations of cells, ethanol,
glycerol and fructose that are the variables of interest
in this bioprocess. To determine them, an open
loop simulation of Eq. (1) was done using the initial
conditions of Table 1, the parameters of Table 2 and
the optimal feed rate policy determined in (Hunag et
al., 2012). Fig. 2: illustrates the obtained reference
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
425
concentration proles, the culture volume and the
corresponding feed ow rate.
To begin with the development of the technique,
as the mathematical model is given as a system of
ordinary differential equations, a numerical method is
used to integrate them. Euler is applied in this study
because it is simple and yields good results.
With this new appreciation of Eq. (1) as a system
of linear equations, a simple possibility for calculating
the control action is available. Besides, it could
be expressed in a matrix form by placing the state
variables as a function of U. In this way, it is more
straightforward to clear the control action:
d
dt T
nn
S
σσσ
=
−
+1
In Eq. (3), σ symbolizes each state variable, σn
is the current value of σ measured from the reactor
(online), and σn+1 is the value of σ in the next
measurement instant. TS is the sampling time; for this
study, its value is 0.1 h, adopted taking into account
the recommendations of (Grifths and Smith, 2006),
ensuring that no important event is neglected, and
that the error is minimized and compensated with the
computational cost. The process lasts 15.7 h.
As the state variables in n+1 are unknown, it is
necessary to nd some way to approximate them. In
this paper, the value at n+1 time for each variable is
adopted by a function of the error at n time (see Eq.
(4)). This allows expressing the unknown variables
(σn+1) as a function of the known ones: σn, references
in n+1 and in n, and a constant.
σσ σσσ
σrefn nref nn
error
n
errornn
k
,,
()
++ +
−= −→
+
11 1
1
==− −
+
σσσ
σrefn refn n
k
,,
()
1
In this equation, σref are the reference state variable
values at the corresponding instant of time. kσ is a
constant that represents the controller parameter for
the variable σ. There are ve different kσ for this model
of ethanol production, one for each variable, kX, kS1,
kS2, kP1 and kP2.
Substituting Eq. (4) in Eq. (3) gives the following
expression that allows the approximation of the
derivatives:
d
dt
k
T
refn refn
nn
S
n
σσσ
σσ
σ
σ
=−−
−
+
+
,,
()
1
1
Substituting Eq. (5) in the mathematical model:
( )
( )
( )
( )
( )
( )
( )
( )
11 12
11 21
22 2,
21
2
1
12 2
ref ,n 1 X ref ,n n n n
1,n 2,n n n
Sn
1ref,n 1 1ref,n 1,n 1,n S/P ,n S /P ,n n
n f 1,n
S P /S P /S n
2ref,n 1 S 2ref,n 2,n 2,n S /P n
S / P ,n
S P /S
S
P /S
X kX X X U
µ µX X
TV
S kS S S qq U
X SS
T YY V
S kS S S q
q
T YY
+
+
+
− −− =+−
− −−
=− + + λ−
− −−
=−+
( )
( )
( )
( )
( )
( )
( )
( )
1
11 21
2
12 22
n
n f 2,n
n
1ref, n 1 P 1ref , n 1n 1,n n
S /P ,n S / P ,n n 1,n
Sn
2ref,n 1 P 2ref,n 2,n 2,n n
S /P ,n S /P ,n n 2,n
Sn
U
X 1 SS
V
P kP P P U
q qX P
TV
P kP P P U
q qX P
TV
+
+
+ −λ −
− −−
=+−
− −−
=+−
( )
( )
( )
( )
( )
( )
( )
( )
111 12
11 21
2
ref, n 1 X ref ,n n n
1,n 2,n n
S
1ref,n 1 1ref,n 1,n 1,n S /P ,n S /P ,n
n
nn
S P /S P /S
f 1,n n
2ref ,n 1 S
f 2,n n n
1,n n
S
2,n n
A
U
X kX X X µ µX
T
S kS S S qq
X
X V T YY
S

/
S /V
Sk
1 S S /V U
P / V
P / V
+
+
+
− −−−+
− −−
++
λ
−
λ=
( )
( )
( )
( )
( )
( )
( )
( )
21 22
12 22
1
11 21
2
12 22
2ref,n 2,n 2,n S /P ,n S /P ,n
n
S P /S P /S
1ref,n 1 P 1ref,n 1,n 1,n
S /P ,n S /P ,n n
S
2ref,n 1 P 2ref,n 2,n 2,n
S /P ,n S /P ,n n
S
b
S SS qq
X
T YY
P kP P P q qX
T
P kP P P q qX
T
+
+
−−
++
− −−−+
− −−−+
To simplify the mathematical expression of the
problem, Eq. (7) is expressed generically as AU=b:
a
a
a
a
a
U
b
b
b
b
b
1
2
3
4
5
1
2
3
4
5
=
To nd U, the system must have an exact solution.
To accomplish this, b has to be a linear combination of
A columns (Strang, 2006), that is to say, A and b must
be parallel. This condition can be satised in different
ways; one of them is that the angle between A and b
must be zero:
cos( ,)
,
  .
Ab
Ab
Ab
=
<>
=1
Here, the operation between < > and . represent
the inner product and the norm of the vectors in the
Rn space, respectively. Consequently, Eq. (9) can be
expressed as:
()ab ab ab ab ab aaaaabbb
11 22 33 44 55 1
2
2
2
3
2
4
2
5
2
1
2
2
2
3
2
++++ −++++ +++bbb
4
2
5
20
+=
For Eq. (10) to have solution, the “sacriced
variable” is dened, which is denoted by the subscript
“ez”. To select it, it is essential to examine and interpret
the role of each variable in the process. In a bioprocess,
the substrate concentration, which can be adjusted by
varying the supply ow rate, directly affects the rate
of substrate consumption, growth rate of cells, and
formation rates of products and byproducts (Öztürk et
al., 2016). As can be appreciated in Eq. (1), the kinetic
model for cell growth and ethanol formation considers
the interactions between two types of sugars: glucose
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
426
and fructose (Hunag et al., 2012). However, few
articles have addressed this issue because the inuence
of glucose is more signicant than that of fructose.
Therefore, S1 is chosen as the sacriced variable.
Then, by replacing S1ref by S1ez in Eq. (7) and applying
the parallelism condition of Eq. (10), the sacriced
variable is readily to be found. Once S1ez,n is calculated,
the control action (Un) can be determined at any sampling
time. To accomplish this, S1ez is replaced in Eq. (7) and Un
is obtained by using least squares (Strang, 2006).
The following procedure aims to nd the best
values for these parameters, between 0 and 1, such
that the tracking error is minimized. So, viewing this
as a problem of searching in an unknown environment,
an appropriate algorithm must be chosen in order to
solve it. Several variants of search problems have
been proposed (e.g., Chrobak et al., 2008). Genetic
Algorithms are widely used to solve complex
optimization problems (Holland, 1975, Sadatsakkak et
al., 2015, Asadi et al., 2014, Yu et al., 2015, Ismail et
al., 2014, Alvarez et al., 2008); this method imitates the
theory of biological evolution proposed by Darwin for
the resolution of problems. Ant Colony Optimization
is a probabilistic technique to solve computational
problems that can be reduced to looking for the best
paths in graphs; it is based on the behavior of ants
when they are looking for a way between the colony
and a food source (Chiha et al., 2012, Castillo et al.,
2015, Omar et al., 2013). The Monte Carlo Algorithm
gives statistical estimates of the required solution by
performing random sampling of a random variable,
whose mathematical expectation is the desired solution
(Dimov et al., 2015).
In this test, the Monte Carlo algorithm is applied
to tune the controller. This method is chosen because
it has several advantages: i) there is low probability
of producing an incorrect result (Motwani and
Raghavan, 1995); ii) less computational complexity in
comparison with other algorithms (Tempo and Ishii,
2007); iii) often, a slight modication of the algorithm
for solving systems of linear algebraic equations
allows one to solve other linear algebra problems
such as matrix inversion and computing the extreme
eigenvalues (Dimov et al., 2015); iv) high reliability
and easy application (Cheein and Scaglia, 2014, de
Oliveira et al., 2012, Mohammadi et al., 2014).
The experiment consists of simulating the process
a number of times while using random values of kσ.
Then, the total error is calculated for each iteration.
The kσ values that make a minimum total error are
selected.
To determine the number of simulations (N), Eq.
(14) is used (Tempo and Ishii, 2007). Note that in order
to limit the chance of a wrong answer, appropriate
condence (δ) and accuracy (ε) must be indicated.
UAAAb
ab ab ab ab ab
aaaaa
TT
=
()
=
++++
++++
−111 22 33 44
55
1
2
2
2
3
2
4
2
5
2
Replacement with component values of each
corresponding matrix:
U
XXkXXX
TµµXV
n
n
refn Xref nn n
S
nnnn
=
−−
()
−
()
−+
()
+
,,
,,
1
12
+
()
+
()
()
+
  

,,,,
,
XSSSSPP
SS
nfnfnnn
f
2
1
2
2
2
1
2
2
2
1
1λλ
λ
nn
ez nezn nn
S
SPn
PS
S
n
S
VSkSSS
T
q
Y
q
()
−−
()
−
()
++
+11 111
111
11
,,,, /,
/
112
21
2
1
2
2
1
/,
/
,
 
Pn
PS
n
nfnf
YX
XSSSS
+
()
+
()
λλ
,, ,,
,
,,


nnn
fn
refn Sref
n
PP
SS SkS
V
()
+
()
()
−
+
2
1
2
2
2
2
21 2
12
λnnn n
S
SPn
PS
SPn
PS
n
SS
T
q
Y
q
YX
−
()
−
()
++
22 21
12
22
22
,, /,
/
/,
/
+
()
+
()
()
+
  
,,,,
XSSSSPP
nfnfnnn
2
1
2
2
2
1
2
2
2
1λλ
 ,
,,,,
/, /
PPkPPP
TqqV
n
refn Pref nn n
S
SPnSn1
11 111
1
11 2
+−−
()
−
()
−+
PPn n
nfnfnn
X
XSSSSPP
1
2
1
2
2
2
1
2
2
1
,
,,,
  
()
+
()
+
()
()
λλ ,,
,
,,,,
/

n
n
refn Pref nn n
S
SPn
PPkPPP
TqV
2
2
21 222
2
1
+
−−
()
−
()
−
+
2222
2
1
2
2
2
1
,/,
,,
  
nSPn n
nfnfn
qX
XSSSS
+
()
+
()
+
()
()
λλPPP
nn1
2
2
2
,,

Equation is the mathematical representation of the
control action required to achieve the concentration
proles shown in Fig. 2.
Note that one of the advantages of this technique
is that it can be applied in many systems, both single
input multipleoutput and multipleinputs multiple
outputs, if the information required is available.
Controller Parameters Selection.
Here, a new term is introduced, the “tracking error”,
dened as follows:
e
XX XP
PP
P
n
refn nref refn
nr
ef
r
=−+−
+
(( )(
()
((
,,
,
)/ma
x)
/max
2
11 1
2
2eef nn refref nn ref
PPSS S
,, ,,
)/ma
x)
/max)(
()−−
+
22
2
22 2
2
where max σre f is the maximum value of the
corresponding reference variable.
As was introduced aboce, the performance of
the bioreactor is directly affected by the controller
parameters (kσ). Those parameters take values between
zero and one (0 < kσ < 1), hence the tracking error
tends to zero when n tends to innity (go to Appendix
A to see the demonstration).
N≥
−
log
log
1
1
1
δ
ε
According to the desired precision of the results, δ
and ε values are designated. The selected values are:
δ=0.01 and ε=0.005. Consequently, N=1000.
(11)
(12)
(13)
(14)
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
427
The total error (Ep) is calculated by adding tracking
errors (en, dened in Eq. (13)) at each sampling
instant n = 1, 2, ..., 157:
where
ˆ
n
x
represents the estimated state variables
of the process, and xn is the offline measured state
(whose components are: Xn, P1n, P2n). The nonlinear
dynamics of the system, described in Eq. (7), can be
represented by an exact neural estimator denoted by:
E=T
ps n
n
e
=
∑
1
157
In the last equation, p = 1, 2, ..., N.
Steps summary:
 Dene δ and ε, (Tempo and Ishii, 2007).
 Calculate N, Eq. (14).
 Compute the tracking error in each time instant,
Eq. (13).
 Obtain the total error at the end of the simulation,
Eq. (15).
 Simulate the process N times.
 Compare all the total errors.
 Select the k values that minimize the total error.
It is noteworthy that the invested time for this
procedure is less than 1 hour. The kσ values that
minimize the tracking are listed in Table 3.
Table 3. Optimal Controller Parameters.
NEURAL STATE ESTIMATION DESIGN
State estimation techniques have a long
development history focused mainly to supply the
lack of system measurements (Salau et al., 2012).
During a fermentation process, variables such as cell
and product concentrations are determined by off
line laboratory analysis, making this set of variables
of limited use for control purposes (de Assis and
Filho, 2000). Although a specic sensor could be
available, this kind of hardware is usually expensive.
According to this, there are many studies for state
estimation in chemical and biochemical processes
(Bogaerts and Coutinho, 2014, Bogaerts and Wouwer,
2003, Dewasme et al., 2015, Hulhoven et al., 2006,
LaraCisneros et al., 2016, Heidarinejad et al., 2012,
Kravaris et al., 2007, Hulhoven et al., 2008, Oliveira
Esquerre et al., 2002, Pantano et al., 2017b).
To implement the proposed control technique, it is
necessary to have a good variable state estimation to
feedback all control variables. Consequently, a neural
network state estimation is developed in order to
provide cells, bioethanol and glycerol concentrations
online. Moreover, the estimator has been trained
in order to obtain good responses against different
perturbations. The estimator design is detailed below.
First it is necessary to dene the estimated state
error
n
x
as follow:
n nn
ˆ
x xx= −
xW
xx
n
T
nn+=
()
+=
()
10
0
*
ξυ ε
where υn is the input vector to the neural estimator, being
υn = [Un, Xn, P1,n, P2,n], and W* ∈ Rm×l is the optimal
weight vector, ε ∈ Rl×1 is the neural approximation
error and ξi is the RBF that represent each neuron in
the hidden layers, subindex i indicates the neuron
number of radial based function (RBF) functions (ξ ∈
Rm×1), l is the number of estimated variables (l=3) and
m is the maximum number of neurons (m=10)
ξυ τυ
in
i
ni
im
()
=− −
=ex
p,
,...
1
212
2
2
c
where ci and τi are parameter vectors of centers and
widths of the RBF respectively.
Since the output state vector is nonmeasured
or affected by disturbances, then there is a need to
estimate the values. A state estimator function based
on Eq. (17) is determined as follows:
( )
T
n1 n n
ˆ
ˆ
xW
+= xu
Now, from the difference between Eq. (17) and Eq.
(19), the neural identication error can be described
by:
( ) ( ) ( )
( )
*T T
n1 n1 n1 n n n n
T
nnn
ˆ
x x x W W k ...
W
+ ++
= − = xu − xu +ε =
= x u +ε
where
n
W
represents the neural weight vector
estimation error and can be dened as:
WWW
nn
=−
*
To train the neural network for identication, an
offline data set of (x, υ) pairs was used. The learning
rule to train the neural will be demonstrated in the next
theorem:
Theorem: Considering the bioethanol process
dened in Eq. (1), it can be approximated by the
neural network Eq. (19) using a neuronal adjustment
law dened by:
( )
T
n nn
Wx∆ = −γx u Λ
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
428
where Λ = diag [λ1, λ2, λ3] is a diagonal denite positive
matrix and γ is an arbitrary positive constant.
Proof: the demonstration of this theorem was added
in Appendix B; the convergence of this estimator is
very important, because an exact representation of
the variables to be estimated (
1, 2,
ˆˆ ˆ
ˆ,,
n nnn
XPP
=
x
) is
necessary.
CONTROLLER PERFORMANCE
EVALUATION
In this section two different tests are carried out
with the intent of demonstrating the reliability of
the technique. Those tests are: adding parametric
uncertainty and perturbations in the control action.
Moreover, the controller is shown while working
under normal operation conditions.
Controller performance under normal conditions.
This simulation is made with the aim of having
a point of comparison when the different tests are
analyzed. To carry it out, a closeloop simulation of
the process is executed by using the information from
Table 1, Table 2, Table 3 and Fig. 2:. It should be noted
that it is being considered that no external disturbances
can affect the process.
Fig. 3 shows the evolution of cells, ethanol, glycerol
and fructose concentration proles during the simulation;
both results with and without estimation are shown.
Those proles are also compared with the references;
note how the state variables follow them perfectly. This
gure also shows the accumulated error and the tracking
error (en). The neural network introduces a limited
intrinsic error (ɛ, neuronal approach error, Eq. (17)), this
error is accumulated with the progress of the process,
the reason why the accumulated error is different for
the results with estimation and noestimation. As can be
seen, as the process moves forward, the tracking error
tends to decrease and remains limited to low values,
i.e., the controller is progressively approaching to the
reference in each instant of time.
Test adding parametric uncertainty.
As is known, a particular feature of bioprocesses is
the difculty of measuring their parameters, especially
because often they vary over time (Wechselberger et
al., 2010). Thus, the next test intends to show how
the controller responds when the values of the system
parameters are not accurate or uctuate throughout the
process.
To fulll what is proposed, the Monte Carlo
algorithm is used. The number of simulations (N) is
determined with Eq. (14), taking into account the same
values for δ and ε. During the N simulations, all system
parameters are randomly changed by +15% of their
original value at the same time (see Table 2).
Fig. 4, shows the concentrations of biomass,
fructose and products throughout the process taking
Figure 3. Comparison between the real cells, ethanol, glycerol and fructose concentrations with the references
under normal operation conditions. Note how the tracking error tends to decrease and remains limited to low values.
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
429
into account the changes in the system parameters.
It is important to highlight that every reference is
almost perfectly followed, which means that this
controller gives excellent results even in the presence
of uncertainties.
Controller performance with perturbations in the
control action.
This test aims to simulate unforeseen events that
may cause unwanted variations in the production. To
carry it out, a random perturbation in the feed rate
of the bioreactor is added. This perturbation affects
the control action by +20% of its original value,
as is revealed in Fig. 5. This can be explained as a
random noise that results in nonzeromean Gaussian
disturbances (George, 2014).
In Fig. 5 the perturbed feed rate prole can be seen in
comparison with the original one. Moreover, this gure
also illustrates how the reference concentrations are
followed to perfection despite the applied disturbance.
Finally, note how this perturbations cause an increase
of the tracking error (with respect to the one analyzed
in Fig. 3), however, it remains at acceptable levels.
CONTROLLER COMPARISON
As was mentioned before, despite the abundance
of sophisticated control techniques for nonlinear
Figure 4. Desired concentration prole variations under parametric uncertainty.
Figure 5. Controller response to the addition of a disturbance in the control action.
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
430
Figure 7. Performance comparison of both controllers with nonzeromean Gaussian disturbances in the control
action.
Figure 6. Comparison of the performance of both controllers under normal conditions.
Figure 8. Performance comparison of both controllers with 30% step perturbation in the control action.
systems, PID or PI controllers are still the most widely
used in factories (Imtiaz et al., 2014) because they
have a simple structure and it is easy to adjust their
parameters in comparison with the complexity of
online implementation, the insufcient accuracy of
online solutions and the computational time required
to nd a solution offered by other controllers (Jin et
al., 2014). Nevertheless, it is wellknown that tracking
variable setpoints is a weakness for those controllers
(Aiba, 1979).
This section shows a comparison between the
proposed controller and a traditional PID controller
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
431
in the same bioethanol production system. The best
PID parameters were selected with a Monte Carlo
algorithm. The technique developed in this study does
not present the disadvantages of the aforementioned
sophisticated controllers, so it can be implemented
in nonlinear systems with minimum requirements. In
addition, its performance is signicantly better than a
PID.
In Fig. 6 a comparison of both controllers’ activity
under normal condition is shown. Note the oscillations
of the PID response and the difference reected in the
accumulated error. In Fig. 7 and Fig. 8, the functioning
of the controllers is manifested in the presence of
white noise and step disturbances in the control action,
respectively. It is remarkable that, in all three cases,
the PID signal saturates, causing a sudden decrease in
the substrate concentration which leads to an abrupt
drop in cell concentration, and consequently in the
product formation. This veries that a PID controller
does not work correctly in nonlinear systems. Note
that this effect does not occur with the linear algebraic
controller, which achieves the tracking of the desired
proles with minimal error.
CONCLUSION
A technique for multivariable system control has
been proposed. It has been tested with a fedbatch
bioprocess of ethanol production. The way used to
calculate the control action allows the minimization of
the tracking error obtained, which tends to zero as the
process progresses.
One relevant contribution of this paper is the Monte
Carlo algorithm application, which enables one to nd
the controller parameters (kσ) successfully and allows
evaluating its performance by adding parametric
uncertainties. Furthermore, the results of the different
tests carried out demonstrate the excellence of the
controller.
Moreover, some advantages of this control law are:
i) the controller is independent of the operating point
because it does not use the linearized model; ii) one
does not need specic knowledge to implement it, one
is able to use it just with some basic algebra concepts;
iii) the form of this technique to represent a system
of nonlinear equations allows the calculation of the
control action from linear equations, which reduces
the mathematical complexity; iv) as it uses discrete
equations, it can be programmed in any computer;
v) it can be used in several kinds of systems, such as
SIMO and MIMO, besides, it cannot be utilized only
for bioprocesses, it could be extended to many elds,
hence it is a promising method; vi) it is a reliable
controller (proved in the simulations section), and its
application is inexpensive and fast.
ACKNOWLEDGMENT
The National Council of Scientic and Technological
Research (CONICET) and the Chemical Engineering
Institute (IIQ) from the National University of San
Juan contributed with the support of this investigation.
REFERENCES
Aiba, S. Review of process control and optimization in
fermentation. Biotechnol Bioeng Symp Ser,. 269
281. (1979).
Ajbar, A.,Ali, E. Study of advanced control of ethanol
production through continuous fermentation.
Journal of King Saud UniversityEngineering
Sciences, 29 (1), 111, (2017). https://doi.
org/10.1016/j.jksues.2015.10.005
Alvarez, V., Larico, R., Ianos, Y.,Aznar, M. Parameter
estimation for VLE calculation by global minimization:
the genetic algorithm. Brazilian Journal of Chemical
Engineering, 25 (2), 409418, (2008). https://doi.
org/10.1590/S010466322008000200018
Andrášik, A., Mészáros, A.,De Azevedo, S. F. On
line tuning of a neural PID controller based on
plant hybrid modeling. Computers & Chemical
Engineering, 28 (8), 14991509, (2004). https://
doi.org/10.1016/j.compchemeng.2003.12.002
Asadi, E., Da Silva, M. G., Antunes, C. H., Dias,
L.,Glicksman, L. Multiobjective optimization for
building retrot: A model using genetic algorithm
and articial neural network and an application.
Energy and Buildings, 81 444456, (2014). https://
doi.org/10.1016/j.enbuild.2014.06.009
Bogaerts, P.,Coutinho, D. Robust nonlinear state
estimation of bioreactors based on H∞ hybrid
observers. Computers & Chemical Engineering,
60 315328, (2014). https://doi.org/10.1016/j.
compchemeng.2013.09.013
Bogaerts, P.,Wouwer, A. V. Software sensors for
bioprocesses. ISA transactions, 42 (4), 547558, (2003).
https://doi.org/10.1016/S00190578(07)600056
Castillo, O., Neyoy, H., Soria, J., Melin, P.,Valdez,
F. A new approach for dynamic fuzzy logic
parameter tuning in ant colony optimization and
its application in fuzzy control of a mobile robot.
Applied soft computing, 28 150159, (2015).
https://doi.org/10.1016/j.asoc.2014.12.002
Chang, L., Liu, X.,Henson, M. A. Nonlinear model
predictive control of fedbatch fermentations
using dynamic ux balance models. Journal of
Process Control, 42 137149, (2016). https://doi.
org/10.1016/j.jprocont.2016.04.012
Cheein, F. A.,Scaglia, G. Trajectory tracking controller
design for unmanned vehicles: A new methodology.
Journal of Field Robotics, 31 (6), 861887, (2014).
https://doi.org/10.1002/rob.21492
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
432
Chiha, I., Liouane, N.,Borne, P. Tuning PID controller
using multiobjective ant colony optimization.
Applied Computational Intelligence and Soft
Computing, 2012 11, (2012). https://doi.
org/10.1155/2012/536326
Chrobak, M., Kenyon, C., Noga, J.,Young, N.
E. Incremental medians via online bidding.
Algorithmica, 50 (4), 455478, (2008). https://doi.
org/10.1007/s004530079005x
Dai, W., Word, D. P.,Hahn, J. Modeling and dynamic
optimization of fuelgrade ethanol fermentation
using fedbatch process. Control Engineering
Practice, 22 231241, (2014). https://doi.
org/10.1016/j.conengprac.2013.01.005
De Assis, A. J.,Filho, R. M. Soft sensors development
for online bioreactor state estimation. Computers &
Chemical Engineering, 24 (2), 10991103, (2000).
https://doi.org/10.1016/S00981354(00)004890
De Battista, H., Picó, J.,PicóMarco, E. Nonlinear
PI control of fedbatch processes for growth
rate regulation. Journal of Process Control, 22
(4), 789797, (2012). https://doi.org/10.1016/j.
jprocont.2012.02.011
De Oliveira, L. P., Verstraete, J. J.,Kolb, M. A
Monte Carlo modeling methodology for the
simulation of hydrotreating processes. Chemical
engineering journal, 207 94102, (2012). https://
doi.org/10.1016/j.cej.2012.05.039
Del RioChanona, E. A., Zhang, D.,Vassiliadis, V.
S. Modelbased realtime optimisation of a fed
batch cyanobacterial hydrogen production process
using economic model predictive control strategy.
Chemical Engineering Science, 142 289298,
(2016). https://doi.org/10.1016/j.ces.2015.11.043
Dewasme, L., Fernandes, S., Amribt, Z., Santos,
L., Bogaerts, P.,Wouwer, A. V. State estimation
and predictive control of fedbatch cultures of
hybridoma cells. Journal of Process Control,
30 5057, (2015). https://doi.org/10.1016/j.
jprocont.2014.12.006
Dimov, I., Maire, S.,Sellier, J. M. A new Walk on
Equations Monte Carlo method for solving systems
of linear algebraic equations. Applied Mathematical
Modelling, 39 (15), 44944510, (2015). https://doi.
org/10.1016/j.apm.2014.12.018
Fu, Y.,Chai, T. Nonlinear multivariable adaptive
control using multiple models and neural networks.
Automatica, 43 (6), 11011110, (2007). https://doi.
org/10.1016/j.automatica.2006.12.010
Fujiki, T. L., Sanzovo, M. R., Leite, M. S., Silva,
F. V.,Fileti, A. M. Optimizing Control Action
Online Using a Neural Model and the Solver of an
Electronic Worksheet. Computer Aided Chemical
Engineering, 27 14611466, (2009). https://doi.
org/10.1016/S15707946(09)706340
George, J. On adaptive loop transfer recovery using
Kalman lterbased disturbance accommodating
control. Control Theory & Applications, IET, 8
(4), 267276, (2014). https://doi.org/10.1049/iet
cta.2013.0671
Grifths, D. V.,Smith, I. M. Numerical methods for
engineers, CRC press. (2006)
Guay, M., Dochain, D.,Perrier, M. Adaptive
extremum seeking control of continuous stirred
tank bioreactors with unknown growth kinetics.
Automatica, 40 (5), 881888, (2004). https://doi.
org/10.1016/j.automatica.2004.01.002
Hecklau, C., Pering, S., Seibel, R., Schnellbaecher, A.,
Wehsling, M., Eichhorn, T., Von Hagen, J.,Zimmer,
A. SSulfocysteine simplies fedbatch processes
and increases the CHO specic productivity via
antioxidant activity. Journal of biotechnology,
218 5363, (2016). https://doi.org/10.1016/j.
jbiotec.2015.11.022
Heidarinejad, M., Liu, J.,Christodes, P. D. State
estimationbased economic model predictive
control of nonlinear systems. Systems & Control
Letters, 61 (9), 926935, (2012). https://doi.
org/10.1016/j.sysconle.2012.06.007
Heux, S., MeynialSalles, I., O’donohue, M.,Dumon,
C. White biotechnology: State of the art
strategies for the development of biocatalysts
for biorening. Biotechnology advances, 33 (8),
16531670, (2015). https://doi.org/10.1016/j.
biotechadv.2015.08.004
Hofmann, S., Bajcinca, N., Raisch, J.,Sundmacher,
K. Optimal control of univariate and multivariate
population balance systems involving external
nes removal. Chemical Engineering Science,
168 101123, (2017). https://doi.org/10.1016/j.
ces.2016.12.032
Holland, J. H. Adaptation in natural and articial
systems. An introductory analysis with application
to biology, control, and articial intelligence. Ann
Arbor, MI: University of Michigan Press, (1975).
Hulhoven, X., Vande Wouwer, A.,Bogaerts, P. State
observer scheme for joint kinetic parameter and
state estimation. Chemical Engineering Science, 63
(19), 48104819, (2008). https://doi.org/10.1016/j.
ces.2007.11.042
Hulhoven, X., Wouwer, A. V.,Bogaerts, P. Hybrid
extended Luenbergerasymptotic observer for
bioprocess state estimation. Chemical Engineering
Science, 61 (21), 71517160, (2006). https://doi.
org/10.1016/j.ces.2006.06.018
Hunag, W.H., Shieh, G. S.,Wang, F.S. Optimization
of fedbatch fermentation using mixture of sugars
to produce ethanol. Journal of the Taiwan Institute
of Chemical Engineers, 43 (1), 18, (2012). https://
doi.org/10.1016/j.jtice.2011.06.007
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
433
Imtiaz, U., Assadzadeh, A., Jamuar, S. S.,Sahu, J. N.
Bioreactor temperature prole controller using
inverse neural network (INN) for production
of ethanol. Journal of Process Control, 23 (5),
731742, (2013). https://doi.org/10.1016/j.
jprocont.2013.03.005
Imtiaz, U., Jamuar, S. S., Sahu, J.,Ganesan, P.
Bioreactor prole control by a nonlinear auto
regressive moving average neuro and two degree
of freedom PID controllers. Journal of Process
Control, 24 (11), 17611777, (2014). https://doi.
org/10.1016/j.jprocont.2014.09.012
Ismail, M., Moghavvemi, M.,Mahlia, T. Genetic
algorithm based optimization on modeling and
design of hybrid renewable energy systems. Energy
Conversion and Management, 85 120130, (2014).
https://doi.org/10.1016/j.enconman.2014.05.064
Jin, H., Chen, X., Yang, J., Wu, L.,Wang, L. Hybrid
intelligent control of substrate feeding for industrial
fedbatch chlortetracycline fermentation process.
ISA transactions, 53 (6), 18221837, (2014). https://
doi.org/10.1016/j.isatra.2014.08.015
Karakuzu, C., Türker, M.,Öztürk, S. Modelling, on
line state estimation and fuzzy control of production
scale fedbatch baker’s yeast fermentation. Control
Engineering Practice, 14 (8), 959974, (2006).
https://doi.org/10.1016/j.conengprac.2005.05.007
Kookos, I. K. Optimization of Batch and Fed
Batch Bioreactors using Simulated Annealing.
Biotechnology progress, 20 (4), 12851288, (2004).
https://doi.org/10.1021/bp034233g
Kravaris, C., Sotiropoulos, V., Georgiou, C., Kazantzis,
N., Xiao, M.,Krener, A. J. Nonlinear observer
design for state and disturbance estimation. Systems
& Control Letters, 56 (11–12), 730735, (2007).
https://doi.org/10.1016/j.sysconle.2007.05.001
LaraCisneros, G., AguilarLópez, R., Dochain,
D.,Femat, R. Online estimation of VFA
concentration in anaerobic digestion via methane
outow rate measurements. Computers &
Chemical Engineering, 94 250256, (2016). https://
doi.org/10.1016/j.compchemeng.2016.07.005
Lehouche, H., Guéguen, H.,Mendil, B. Supervisory
control based on closed‐loop adaptive control
approach of nonlinear systems: application to
CSTR process. Asian Journal of Control, 14 (1),
258270, (2012). https://doi.org/10.1002/asjc.317
Liu, C., Gong, Z., Shen, B.,Feng, E. Modelling and
optimal control for a fedbatch fermentation
process. Applied Mathematical Modelling, 37
(3), 695706, (2013). https://doi.org/10.1016/j.
apm.2012.02.044
Logist, F., Houska, B., Diehl, M.,Van Impe, J. F.
A Toolkit for MultiObjective Optimal Control
in Bioprocess Engineering. IFAC Proceedings
Volumes, 43 (6), 281286, (2010). https://doi.
org/10.3182/201007073BE2012.0063
Mangesh, M. G.,Jana, A. K. A comparison of three sets
of DSP algorithms for monitoring the production
of ethanol in a fedbatch baker’s yeast fermenter.
Measurement, 41 (9), 970985, (2008). https://doi.
org/10.1016/j.measurement.2008.01.008
Mohammadi, Y., Pakdel, A. S., Saeb, M. R.,Boodhoo,
K. Monte Carlo simulation of free radical
polymerization of styrene in a spinning disc reactor.
Chemical Engineering Journal, 247 231240,
(2014). https://doi.org/10.1016/j.cej.2014.03.013
Mohammadzaheri, M.,Chen, L. Intelligent control of
a nonlinear tank reactor. Asian Journal of Control,
13 (3), 439444, (2011). https://doi.org/10.1002/
asjc.347
Motwani, R., Raghavan, P. . Randomized algorithms
(cambridge international series on parallel
computation). Cambridge University Press, (1995).
https://doi.org/10.1017/CBO9780511814075
Ochoa, S. A new approach for nding smooth optimal
feeding proles in fedbatch fermentations.
Biochemical Engineering Journal, 105 177188,
(2016). https://doi.org/10.1016/j.bej.2015.09.004
OliveiraEsquerre, K., Mori, M.,Bruns, R. Simulation
of an industrial wastewater treatment plant using
articial neural networks and principal components
analysis. Brazilian Journal of Chemical
Engineering, 19 (4), 365370, (2002). https://doi.
org/10.1590/S010466322002000400002
Omar, M., Soliman, M., Ghany, A. A.,Bendary, F.
Optimal tuning of PID controllers for hydrothermal
load frequency control using ant colony
optimization. International journal on electrical
engineering and informatics, 5 (3), 348, (2013).
https://doi.org/10.15676/ijeei.2013.5.3.8
Öztürk, S., Çalık, P.,Özdamar, T. H. FedBatch
Biomolecule Production by Bacillus subtilis: A State
of the Art Review. Trends in Biotechnology, (2016).
https://doi.org/10.1016/j.tibtech.2015.12.008
Pantano, M. N., Fernández, M. C., Serrano, M. E.,
Ortíz, O. A.,Scaglia, G. J. E. Trajectory Tracking
Controller for a Nonlinear Fedbatch Bioprocess.
Revista Ingeniería Electrónica, Automática y
Comunicaciones, 38 (1), 78, (2017a).
Pantano, M. N., Serrano, M. E., Fernández, M. C.,
Rossomando, F. G., Ortiz, O. A.,Scaglia, G. J.
Multivariable Control for Tracking Optimal Proles
in a Nonlinear FedBatch Bioprocess Integrated
with State Estimation. Industrial & Engineering
Chemistry Research, 56 (20), 60436056, (2017b).
https://doi.org/10.1021/acs.iecr.7b00831
Preuβ, K., Le Lann, M.V., Cabassud, M.,Anne
Archard, G. Implementation procedure of an
advanced supervisory and control strategy in the
pharmaceutical industry. Control engineering
practice, 11 (12), 14491458, (2003). https://doi.
org/10.1016/S09670661(03)000984
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
434
Roeva, O.,Slavov, T. PID Controller Tuning based on
metaheuristic algorithms for bioprocess control.
Biotechnology & Biotechnological Equipment, 26
(5), 32673277, (2012). https://doi.org/10.5504/
BBEQ.2012.0065
Sadatsakkak, S. A., Ahmadi, M. H., Bayat, R.,
Pourkiaei, S. M.,Feidt, M. Optimization density
power and thermal efciency of an endoreversible
Braysson cycle by using nondominated sorting
genetic algorithm. Energy Conversion and
Management, 93 3139, (2015). https://doi.
org/10.1016/j.enconman.2014.12.088
Salau, N. P., Trierweiler, J. O.,Secchi, A. R.. State
estimators for better bioprocesses operation.
Computer Aided Chemical Engineering. Elsevier.
(2012). https://doi.org/10.1016/B9780444
595201.501123
Scaglia, G., Rosales, A., Quintero, L., Mut, V., Agarwal,
R. A linearinterpolationbased controller design
for trajectory tracking of mobile robots. Control
Engineering Practice, 18 (3), 318329, (2010).
https://doi.org/10.1016/j.conengprac.2009.11.011
Strang, G. Linear algebra and its applications,
Thomson, Brooks/Cole, Belmont, CA. ISBN
0030105676. (2006)
Tempo, R.,Ishii, H. Monte Carlo and Las Vegas
Randomized Algorithms for Systems and Control*:
An Introduction. European journal of control, 13
(2), 189203, (2007). https://doi.org/10.3166/
ejc.13.189203
Wechselberger, P., Seifert, A.,Herwig, C. PAT method
to gather bioprocess parameters in realtime
using simple input variables and rst principle
relationships. Chemical Engineering Science, 65
(21), 57345746, (2010). https://doi.org/10.1016/j.
ces.2010.05.002
Wood, B. J. Microbiology of fermented foods, Springer
Science & Business Media. (2012)
Xu, L., Ding, F., Gu, Y., Alsaedi, A.,Hayat, T. A
multiinnovation state and parameter estimation
algorithm for a state space system with dstep
statedelay. Signal Processing, (2017). https://doi.
org/10.1016/j.sigpro.2017.05.006
Ye, J., Xu, H., Feng, E.,Xiu, Z. Optimization of a fed
batch bioreactor for 1, 3propanediol production
using hybrid nonlinear optimal control. Journal
of Process Control, 24 (10), 15561569, (2014).
https://doi.org/10.1016/j.jprocont.2014.08.002
Yu, W., Li, B., Jia, H., Zhang, M.,Wang, D. Application
of multiobjective genetic algorithm to optimize
energy efciency and thermal comfort in building
design. Energy and Buildings, 88 135143, (2015).
https://doi.org/10.1016/j.enbuild.2014.11.063
Zhou, K., Doyle, J.,Glover, K. Robust and Optimal
Control, Prentice Hall. (1996)
APPENDIX A
This appendix aims to demonstrate the error
convergence to zero. As was explained in the article,
the values of the kσ parameters must be between zero
and one (0 < kσ < 1), because that makes the tracking
error tends to zero when n tends to innity. The steps
to demonstrate this fact are:
Once the sacriced variable is determined,
the matrix form of the equations system could be
generically expressed as:
a
a
a
a
a
U
b
b
b
b
b
A
n
b
1
2
3
4
5
1
2
3
4
5
=
Then, solving Eq. (A.1) with least squares:
UAAAb
ab ab ab ab ab
aaaaa
n
TT
=
()
=
++++
++++
−111 22 33 44
55
1
2
2
2
3
2
4
2
5
2
From Eq. (A.1):
a
a
b
bb
a
ab
a
a
b
bba
ab
a
a
b
bba
ab
a
1
2
1
2
2
2
1
1
1
3
1
3
3
3
1
1
1
4
1
4
4
4
1
1
=→=
=→=
=→=
11
5
1
5
5
5
1
1
a
b
bba
ab=→=
Substituting Eq. (A.3) in Eq. (A.2):
Uab ab aaba ab aaba
aa
n=++++
+
11 2
2
11 3
2
11 4
2
11 5
2
11
1
2
2
2
()/()/ ()/()/
++++
=++++
+++
aaa
baaaaaa
aaa
3
2
4
2
5
2
111
2
2
2
3
2
4
2
5
2
1
2
2
2
3
2
(/)( )
aaa
b
a
VX kX XXTS
nrefnXrefn nnS
4
2
5
2
1
1
11
+
−−−−
+
=
= ((( ())/ )((µ111 22 1ezn nn
nn
n
PS
PX
X
,) (,)) )+
−
µ
Replacing Eq. (A.4) in Eq. (7):
XX kX XTSP SP X
nrefnXrefn nS ezn nnnn
++
=− −+ −
11 11 1111
()[( ,) (,)]µµ
Then, as it was mentioned, the tracking error
denition is:
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
State Estimation and Trajectory Tracking Control for a Nonlinear and Multivariable Bioethanol Production System
Brazilian Journal of Chemical Engineering, Vol. 36, No. 01, pp. 421  437, January  March, 2019
435
The same procedure is followed for S2 and the
result is:
eXX
Xn refn n
++
+
=−
11
1
Introducing Eq. (A.5) in Eq. (a.6):
ekXXTSPSPX
Xn XrefnnSezn nnnn
+
=−−−
1111 11 1
()[( ,) (,)]µµ
The Taylor approximation of µ1(S1n, P1n) in the
desired value µ1(S1ez n, P1n) is:
µµ µ
11 1111
111
1
11
1
(,)(,) (, )
()
SP SP dSP
dS SS
nn ezn n
n
nezn
SS
ezn
=+ −
=++−=
→<<
θθ
θ
()
SnSezn S
where
11
01
Replacing Eq. (A.8) in Eq. (A.7):
ekXXTSPdSP
dS SS
Xn Xrefnn Sezn n
n
S
n+=−−+ −
1111
111
1
11
()(,)(, )(µµ
θ
eezn
e
ezn
nn
Xn Xrefnn
Sn
SPX
ekXX
)(,)
()
−
+
−
=−+
1
11 1
1
µ
TT dSP
dS eX
S
n
S
Sn
n
µ
θ
111
1
1
(, )
In the same way, it is demonstrated for S1:
From Eq. (A.1):
a
a
b
bb
a
ab
a
a
b
bba
ab
a
a
b
bba
ab
a
1
2
1
2
1
1
2
2
3
2
3
2
3
3
2
2
4
2
4
2
4
4
2
2
=→=
=→=
=→=
55
2
5
2
5
5
2
2
a
b
bba
ab=→=
Substituting Eq. (A.10) in Eq. (A.2):
Uab aababa ab aaba
aa
n=++++
+
()/()/ ()/()/
1
2
2222 3
2
22 4
2
22 5
2
22
1
2
2
2++++
=++++
+++
aaa
baaaaaa
aaa
3
2
4
2
5
2
221
2
2
2
3
2
4
2
5
2
1
2
2
2
3
2
(/)( )
aaa
b
a
VS kS SSTq
nezn Sezn nnSS
4
2
5
2
2
2
11 111
1
+
−−−+
+
=
= ((( ())/ )111 11 12 21
11 12
1
//
//
(,)/ (, )/ )
PnnPSSPnnP
Sn
fn
SP YqSP
YX
SS
+
()
−λ
Replacing Eq. (A.11) in Eq. (A.7):
SS kS S
nezn Sezn n11 11
11
1
++
=− −()
Then, the tracking error denition is:
eS S
Sref
nn
n11 11
11
+
=−
++
Introducing Eq. (A.12) in Eq. (A.13):
eS SkSS
ekSS
Sn ezn ezn Sezn n
Sn Sezn
11
11
1111111
111
++ +
+
=− +−
=−
(())
(nnSSn
ke)=11
ekSSke
Sn Srefnn
SS
n
22
22
122+
=−=()
In the same way, it is demonstrated for P1:
From Eq. (A.1):
a
a
b
bb
a
ab
a
a
b
bba
ab
a
a
b
bba
ab
a
1
4
1
4
1
1
4
4
2
4
2
4
2
2
4
4
3
4
3
4
3
3
4
4
=→=
=→=
=→=
55
4
5
4
5
5
4
4
a
b
bba
ab=→=
Substituting Eq. (A.16) in Eq. (A.2):
Uab aaba ab aababa
aa
n=++++
+
()/()/ ()/()/
1
2
44 2
2
44 3
2
4444 5
2
44
1
2
2
2++++
=++++
+++
aaa
baaaaaa
aaa
3
2
4
2
5
2
441
2
2
2
3
2
4
2
5
2
1
2
2
2
3
2
(/)( )
aaa
b
a
VP kP PP
nref nPrefn nn
4
2
5
2
4
4
11 111
1
+
−−−
+
=
= ((( ())/
,,,,
TTq SP qS
PX
X
SSPn ez nn SP nn n
n
)( (,)(,)
))
/, ,, /,,
−+
−
11 21
11 21
Substituting Eq. (A.17) in Eq. (7):
PP kP PTqSP
nref nPref nnSSPnezn n11 11 11 11
111
,, ,, /, ,,
()[(,
++
=− −+ ))(,)]
/, ,,
−qSPX
SPnn
nn
11
11
Then, the tracking error denition is:
eP P
Pn refn n
1
11
11
1,,,
++
+
=−
Substituting Eq. (A.18) in Eq. (A.19):
ekPPTq SP q
Pn Pref nnSSPn ez nn SP
11 11 11
11111,,,/,,,/,
()[(,)
+
=−−−
nnn
nn
SP X(,)]
,,11
The Taylor approximation of qS1/P1 (S1n, P1n) in the
desired value qS1/P1 (S1ezn, P1n) is:
qSPq SP qSPd
SPnnnSPn ez nn
SPn
11 11
11
11 11
11
/, ,, /, ,,
/, ,
(,)(,) (,
=+
nn
nezn
e
dS SS
wher
Sn
SS
ez nSnSez nS
)
()
,(,,
)
1
11
1
11 11
−
−
=+
−=
ψψ
ee →< <01ψ
Substituting Eq. (A.21) in Eq. (A.20):
(A.6)
(A.14)
(A.7)
(A.15)
(A.8)
(A.16)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.17)
(A.18)
(A.19)
(A.20)
(A.21)
M. Cecilia Fernández et al.
Brazilian Journal of Chemical Engineering
436
The demonstration is similar for P2 and the result
is:
ekPPTq SP qS
Pn Pref nnSSPn ez nn SP
11 11 11
11111,,,/,,,/
() (,)(
+
=−−−
111
11
1
1
11
1
1
ez nn
SPnn
Sn n
Pn
PqSPd
dS eX
e
S
,,
/, ,
,
,
,) (, )
+
+
ψ
==−−kP PT
d
dS eX
qSP
Pref nnSSnn
SPnn
S
1 1
11
11
1
11
() (, )
,, ,
/, ,
ψ
ekPPT
d
dS
eX
qSP
Pn Pref nn
SSnn
SPnn
S
22 1
12
122
1
12
,,
,,
/, ,
()
(, )
+=−−
α
wwhere→< <01α
Finally, joining Eq. (A.9), (A.14), A.15), (A.22)
and (A.23):
e
e
e
e
e
k
Xn
Sn
Sn
Pn
Pn
X+
+
+
+
+
=
1
1
1
1
1
1
2
1
2
0000
,<